Electromagnetic Modeling within a Microscopically Realistic Brain - Implications for Brain Stimulation.

Across all domains of brain stimulation (neuromodulation), conventional analysis of neuron activation involves two discrete steps: i) prediction of macroscopic electric field, ignoring presence of cells and; ii) prediction of cell activation from tissue electric fields. The first step assumes that current flow is not distorted by the dense tortuous network of cell structures. The deficiencies of this assumption have long been recognized, but - except for trivial geometries - ignored, because it presented intractable computation hurdles. This study introduces a novel approach for analyzing electric fields within a microscopically realistic brain volume. Our pipeline overcomes the technical intractability that prevented such analysis while also showing significant implications for brain stimulation. Contrary to the standard finite element method (FEM), we suggest using a nested iterative boundary element method (BEM) coupled with the fast multipole method (FMM). The nested iterative BEM-FMM approach allows for solving problems with multiple length scales efficiently. It could potentially handle any number of electromagnetically coupled closely spaced neurons and blood microcapillaries. A target application is a subvolume of the L2/3 P36 mouse primary visual cortex containing approximately 400 detailed neuronal meshes at a resolution of 100 nm, which is obtained from scanning electron microscopy data. Our immediate result is a reduction of the stimulation field strength necessary for neuron activation by a factor of 0.85-0.55 (by 15%-45%) as compared to macroscopic predictions. This is in line with experimental data stating that existing macroscopic theories substantially overestimate electric field levels necessary for brain stimulation.

An adaptive h-refinement method for the boundary element fast multipole method for quasi-static electromagnetic modeling.

Objective.In our recent work pertinent to modeling of brain stimulation and neurophysiological recordings, substantial modeling errors in the computed electric field and potential have sometimes been observed for standard multi-compartment head models. The goal of this study is to quantify those errors and, further, eliminate them through an adaptive mesh refinement (AMR) algorithm. The study concentrates on transcranial magnetic stimulation (TMS), transcranial electrical stimulation (TES), and electroencephalography (EEG) forward problems.Approach.We propose, describe, and systematically investigate an AMR method using the boundary element method with fast multipole acceleration (BEM-FMM) as the base numerical solver. The goal is to efficiently allocate additional unknowns to critical areas of the model, where they will best improve solution accuracy. The implemented AMR method's accuracy improvement is measured on head models constructed from 16 Human Connectome Project subjects under problem classes of TES, TMS, and EEG. Errors are computed between three solutions: an initial non-adaptive solution, a solution found after applying AMR with a conservative refinement rate, and a 'silver-standard' solution found by subsequent 4:1 global refinement of the adaptively-refined model.Main results.Excellent agreement is shown between the adaptively-refined and silver-standard solutions for standard head models. AMR is found to be vital for accurate modeling of TES and EEG forward problems for standard models: an increase of less than 25% (on average) in number of mesh elements for these problems, efficiently allocated by AMR, exposes electric field/potential errors exceeding 60% (on average) in the solution for the unrefined models.Significance.This error has especially important implications for TES dosing prediction-where the stimulation strength plays a central role-and for EEG lead fields. Though the specific form of the AMR method described here is implemented for the BEM-FMM, we expect that AMR is applicable and even required for accurate electromagnetic simulations by other numerical modeling packages as well.

Modeling the effects of transcranial magnetic stimulation on spatial attention.

Objectives. Transcranial magnetic stimulation (TMS) has been widely used to modulate brain activity in healthy and diseased brains, but the underlying mechanisms are not fully understood. Previous research leveraged biophysical modeling of the induced electric field (E-field) to map causal structure-function relationships in the primary motor cortex. This study aims at transferring this localization approach to spatial attention, which helps to understand the TMS effects on cognitive functions, and may ultimately optimize stimulation schemes.Approach. Thirty right-handed healthy participants underwent a functional magnetic imaging (fMRI) experiment, and seventeen of them participated in a TMS experiment. The individual fMRI activation peak within the right inferior parietal lobule (rIPL) during a Posner-like attention task defined the center target for TMS. Thereafter, participants underwent 500 Posner task trials. During each trial, a 5-pulse burst of 10 Hz repetitive TMS (rTMS) was given over the rIPL to modulate attentional processing. The TMS-induced E-fields for every cortical target were correlated with the behavioral modulation to identify relevant cortical regions for attentional orientation and reorientation.Main results. We did not observe a robust correlation between E-field strength and behavioral outcomes, highlighting the challenges of transferring the localization method to cognitive functions with high neural response variability and complex network interactions. Nevertheless, TMS selectively inhibited attentional reorienting in five out of seventeen subjects, resulting in task-specific behavioral impairments. The BOLD-measured neuronal activity and TMS-evoked neuronal effects showed different patterns, which emphasizes the principal distinction between the neural activity being correlated with (or maybe even caused by) particular paradigms, and the activity of neural populations exerting a causal influence on the behavioral outcome.Significance. This study is the first to explore the mechanisms of TMS-induced attentional modulation through electrical field modeling. Our findings highlight the complexity of cognitive functions and provide a basis for optimizing attentional stimulation protocols.

An Adaptive H-Refinement Method for the Boundary Element Fast Multipole Method for Quasi-static Electromagnetic Modeling.

Objective: In our recent work pertinent to modeling of brain stimulation and neurophysiological recordings, substantial modeling errors in the computed electric field and potential have sometimes been observed for standard multi-compartment head models. The goal of this study is to quantify those errors and, further, eliminate them through an adaptive mesh refinement (AMR) algorithm. The study concentrates on transcranial magnetic stimulation (TMS), transcranial electrical stimulation (TES), and electroencephalography (EEG) forward problems. Approach: We propose, describe, and systematically investigate an AMR method using the Boundary Element Method with Fast Multipole Acceleration (BEM-FMM) as the base numerical solver. The goal is to efficiently allocate additional unknowns to critical areas of the model, where they will best improve solution accuracy.The implemented AMR method's accuracy improvement is measured on head models constructed from 16 Human Connectome Project subjects under problem classes of TES, TMS, and EEG. Errors are computed between three solutions: an initial non-adaptive solution, a solution found after applying AMR with a conservative refinement rate, and a "silver-standard" solution found by subsequent 4:1 global refinement of the adaptively-refined model. Main Results: Excellent agreement is shown between the adaptively-refined and silver-standard solutions for standard head models. AMR is found to be vital for accurate modeling of TES and EEG forward problems for standard models: an increase of less than 25% (on average) in number of mesh elements for these problems, efficiently allocated by AMR, exposes electric field/potential errors exceeding 60% (on average) in the solution for the unrefined models. Significance: This error has especially important implications for TES dosing prediction - where the stimulation strength plays a central role - and for EEG lead fields. Though the specific form of the AMR method described here is implemented for the BEM-FMM, we expect that AMR is applicable and even required for accurate electromagnetic simulations by other numerical modeling packages as well.

Precise motor mapping with transcranial magnetic stimulation.

We describe a routine to precisely localize cortical muscle representations within the primary motor cortex with transcranial magnetic stimulation (TMS) based on the functional relation between induced electric fields at the cortical level and peripheral muscle activation (motor-evoked potentials; MEPs). Besides providing insights into structure-function relationships, this routine lays the foundation for TMS dosing metrics based on subject-specific cortical electric field thresholds. MEPs for different coil positions and orientations are combined with electric field modeling, exploiting the causal nature of neuronal activation to pinpoint the cortical origin of the MEPs. This involves constructing an individual head model using magnetic resonance imaging, recording MEPs via electromyography during TMS and computing the induced electric fields with numerical modeling. The cortical muscle representations are determined by relating the TMS-induced electric fields to the MEP amplitudes. Subsequently, the coil position to optimally stimulate the origin of the identified cortical MEP can be determined by numerical modeling. The protocol requires 2 h of manual preparation, 10 h for the automated head model construction, one TMS session lasting 2 h, 12 h of computational postprocessing and an optional second TMS session lasting 30 min. A basic level of computer science expertise and standard TMS neuronavigation equipment suffices to perform the protocol.

Comparison of the performance and reliability between improved sampling strategies for polynomial chaos expansion.

As uncertainty and sensitivity analysis of complex models grows ever more important, the difficulty of their timely realizations highlights a need for more efficient numerical operations. Non-intrusive Polynomial Chaos methods are highly efficient and accurate methods of mapping input-output relationships to investigate complex models. There is substantial potential to increase the efficacy of the method regarding the selected sampling scheme. We examine state-of-the-art sampling schemes categorized in space-filling-optimal designs such as Latin Hypercube sampling and L1-optimal sampling and compare their empirical performance against standard random sampling. The analysis was performed in the context of L1 minimization using the least-angle regression algorithm to fit the GPCE regression models. Due to the random nature of the sampling schemes, we compared different sampling approaches using statistical stability measures and evaluated the success rates to construct a surrogate model with relative errors of <0.1%, <1%, and <10%, respectively. The sampling schemes are thoroughly investigated by evaluating the y of surrogate models constructed for various distinct test cases, which represent different problem classes covering low, medium and high dimensional problems. Finally, the sampling schemes are tested on an application example to estimate the sensitivity of the self-impedance of a probe that is used to measure the impedance of biological tissues at different frequencies. We observed strong differences in the convergence properties of the methods between the analyzed test functions.

The influence of white matter lesions on the electric field in transcranial electric stimulation.

BACKGROUND: Transcranial direct current stimulation (tDCS) is a promising tool to enhance therapeutic efforts, for instance, after a stroke. The achieved stimulation effects exhibit high inter-subject variability, primarily driven by perturbations of the induced electric field (EF). Differences are further elevated in the aging brain due to anatomical changes such as atrophy or lesions. Informing tDCS protocols by computer-based, individualized EF simulations is a suggested measure to mitigate this variability. OBJECTIVE: While brain anatomy in general and specifically atrophy as well as stroke lesions are deemed influential on the EF in simulation studies, the influence of the uncertainty in the change of the electrical properties of the white matter due to white matter lesions (WMLs) has not been quantified yet. METHODS: A group simulation study with 88 subjects assigned into four groups of increasing lesion load was conducted. Due to the lack of information about the electrical conductivity of WMLs, an uncertainty analysis was employed to quantify the variability in the simulation when choosing an arbitrary conductivity value for the lesioned tissue. RESULTS: The contribution of WMLs to the EF variance was on average only one tenth to one thousandth of the contribution of the other modeled tissues. While the contribution of the WMLs significantly increased (p≪.01) in subjects exhibiting a high lesion load compared to low lesion load subjects, typically by a factor of 10 and above, the total variance of the EF didnot change with the lesion load. CONCLUSION: Our results suggest that WMLs do not perturb the EF globally and can thus be omitted when modeling subjects with low to medium lesion load. However, for high lesion load subjects, the omission of WMLs may yield less robust local EF estimations in the vicinity of the lesioned tissue. Our results contribute to the efforts of accurate modeling of tDCS for treatment planning.

The effect of meninges on the electric fields in TES and TMS. Numerical modeling with adaptive mesh refinement.

BACKGROUND: When modeling transcranial electrical stimulation (TES) and transcranial magnetic stimulation (TMS) in the brain, the meninges - dura, arachnoid, and pia mater - are often neglected due to high computational costs. OBJECTIVE: We investigate the impact of the meningeal layers on the cortical electric field in TES and TMS while considering the headreco segmentation as the base model. METHOD: We use T1/T2 MRI data from 16 subjects and apply the boundary element fast multipole method with adaptive mesh refinement, which enables us to accurately solve this problem and establish method convergence at reasonable computational cost. We compare electric fields in the presence and absence of various meninges for two brain areas (M1HAND and DLPFC) and for several distinct TES and TMS setups. RESULTS: Maximum electric fields in the cortex for focal TES consistently increase by approximately 30% on average when the meninges are present in the CSF volume. Their effect on the maximum field can be emulated by reducing the CSF conductivity from 1.65 S/m to approximately 0.85 S/m. In stark contrast to that, the TMS electric fields in the cortex are only weakly affected by the meningeal layers and slightly (∼6%) decrease on average when the meninges are included. CONCLUSION: Our results quantify the influence of the meninges on the cortical TES and TMS electric fields. Both focal TES and TMS results are very consistent. The focal TES results are also in a good agreement with a prior relevant study. The solver and the mesh generator for the meningeal layers (compatible with SimNIBS) are available online.

Efficient high-resolution TMS mapping of the human motor cortex by nonlinear regression.

Transcranial magnetic stimulation (TMS) is a powerful tool to investigate causal structure-function relationships in the human brain. However, a precise delineation of the effectively stimulated neuronal populations is notoriously impeded by the widespread and complex distribution of the induced electric field. Here, we propose a method that allows rapid and feasible cortical localization at the individual subject level. The functional relationship between electric field and behavioral effect is quantified by combining experimental data with numerically modeled fields to identify the cortical origin of the modulated effect. Motor evoked potentials (MEPs) from three finger muscles were recorded for a set of random stimulations around the primary motor area. All induced electric fields were nonlinearly regressed against the elicited MEPs to identify their cortical origin. We could distinguish cortical muscle representation with high spatial resolution and localized them primarily on the crowns and rims of the precentral gyrus. A post-hoc analysis revealed exponential convergence of the method with the number of stimulations, yielding a minimum of about 180 random stimulations to obtain stable results. Establishing a functional link between the modulated effect and the underlying mode of action, the induced electric field, is a fundamental step to fully exploit the potential of TMS. In contrast to previous approaches, the presented protocol is particularly easy to implement, fast to apply, and very robust due to the random coil positioning and therefore is suitable for practical and clinical applications.

Boundary element fast multipole method for modeling electrical brain stimulation with voltage and current electrodes.

Objective. To formulate, validate, and apply an alternative to the finite element method (FEM) high-resolution modeling technique for electrical brain stimulation-the boundary element fast multipole method (BEM-FMM). To include practical electrode models for both surface and embedded electrodes.Approach. Integral equations of the boundary element method in terms of surface charge density are combined with a general-purpose fast multipole method and are expanded for voltage, shunt, current, and floating electrodes. The solution of coupled and properly weighted/preconditioned integral equations is accompanied by enforcing global conservation laws: charge conservation law and Kirchhoff's current law.Main results.A sub-percent accuracy is reported as compared to the analytical solutions and simple validation geometries. Comparison to FEM considering realistic head models resulted in relative differences of the electric field magnitude in the range of 3%-6% or less. Quantities that contain higher order spatial derivatives, such as the activating function, are determined with a higher accuracy and a faster speed as compared to the FEM. The method can be easily combined with existing head modeling pipelines such as headreco or mri2mesh.Significance.The BEM-FMM does not rely on a volumetric mesh and is therefore particularly suitable for modeling some mesoscale problems with submillimeter (and possibly finer) resolution with high accuracy at moderate computational cost. Utilizing Helmholtz reciprocity principle makes it possible to expand the method to a solution of EEG forward problems with a very large number of cortical dipoles.

Left posterior inferior parietal cortex causally supports the retrieval of action knowledge.

Conceptual knowledge is central to human cognition. The left posterior inferior parietal lobe (pIPL) is implicated by neuroimaging studies as a multimodal hub representing conceptual knowledge related to various perceptual-motor modalities. However, the causal role of left pIPL in conceptual processing remains unclear. Here, we transiently disrupted left pIPL function with transcranial magnetic stimulation (TMS) to probe its causal relevance for the retrieval of action and sound knowledge. We compared effective TMS over left pIPL with sham TMS, while healthy participants performed three different tasks-lexical decision, action judgment, and sound judgment-on words with a high or low association to actions and sounds. We found that pIPL-TMS selectively impaired action judgments on low sound-low action words. For the first time, we directly related computational simulations of the TMS-induced electrical field to behavioral performance, which revealed that stronger stimulation of left pIPL is associated with worse performance for action but not sound judgments. These results indicate that left pIPL causally supports conceptual processing when action knowledge is task-relevant and cannot be compensated by sound knowledge. Our findings suggest that left pIPL is specialized for the retrieval of action knowledge, challenging the view of left pIPL as a multimodal conceptual hub.

A novel approach to localize cortical TMS effects.

Despite the widespread use of transcranial magnetic stimulation (TMS), the precise cortical locations underlying the resulting physiological and behavioral effects are still only coarsely known. To date, mapping strategies have relied on projection approaches (often termed "center of gravity" approaches) or maximum electric field value evaluation, and therefore localize the stimulated cortical site only approximately and indirectly. Focusing on the motor cortex, we present and validate a novel method to reliably determine the effectively stimulated cortical site at the individual subject level. The approach combines measurements of motor evoked potentials (MEPs) at different coil positions and orientations with numerical modeling of induced electric fields. We identify sharply bounded cortical areas, around the gyral crowns and rims of the motor hand area, as the origin of MEPs and show that the magnitude of the tangential component and the overall magnitude of the electric field are most relevant for the observed effect. To validate our approach, we identified the coil location and orientation that produces the maximal electric field at the predicted stimulation site, and then experimentally show that this location produces MEPs more efficiently than other tested locations/orientations. Moreover, we used extensive uncertainty and sensitivity analyses to verify the robustness of the method and identify the most critical model parameters. Our generic approach improves the localization of the cortical area effectively stimulated by TMS and may be transferred to other modalities such as language mapping.

A principled approach to conductivity uncertainty analysis in electric field calculations.

Uncertainty surrounding ohmic tissue conductivity impedes accurate calculation of the electric fields generated by non-invasive brain stimulation. We present an efficient and generic technique for uncertainty and sensitivity analyses, which quantifies the reliability of field estimates and identifies the most influential parameters. For this purpose, we employ a non-intrusive generalized polynomial chaos expansion to compactly approximate the multidimensional dependency of the field on the conductivities. We demonstrate that the proposed pipeline yields detailed insight into the uncertainty of field estimates for transcranial magnetic stimulation (TMS) and transcranial direct current stimulation (tDCS), identifies the most relevant tissue conductivities, and highlights characteristic differences between stimulation methods. Specifically, we test the influence of conductivity variations on (i) the magnitude of the electric field generated at each gray matter location, (ii) its normal component relative to the cortical sheet, (iii) its overall magnitude (indexed by the 98th percentile), and (iv) its overall spatial distribution. We show that TMS fields are generally less affected by conductivity variations than tDCS fields. For both TMS and tDCS, conductivity uncertainty causes much higher uncertainty in the magnitude as compared to the direction and overall spatial distribution of the electric field. Whereas the TMS fields were predominantly influenced by gray and white matter conductivity, the tDCS fields were additionally dependent on skull and scalp conductivities. Comprehensive uncertainty analyses of complex systems achieved by the proposed technique are not possible with classical methods, such as Monte Carlo sampling, without extreme computational effort. In addition, our method has the advantages of directly yielding interpretable and intuitive output metrics and of being easily adaptable to new problems.